Worksheet 4: Continuity and Derivatives with Limits



Final Review Part 1

1. Prove [pic]

2. Prove [pic] using the delta-epsilon definition of a limit

3. Write the delta-epsilon definition for a limit not existing.

4. An example of a function where the integral does not exist:

[pic]

5. A function that has no antiderivative: [pic]

6. Suppose the function [pic] is defined as [pic] when [pic]. If [pic] exists, find[pic] and [pic]

7. Find the domain, range, and derivative of[pic]

8. Find the derivative and second derivative of[pic]

9. Find the 1000th and 2000th derivatives of[pic]

10. Find the nth derivative of[pic]

11. True or false:

a. [pic]

b. [pic]

c. If [pic]and [pic], then [pic]

d. [pic], [pic] and [pic] implies that [pic] has an inflection point at x = 0

e. L’Hospital’s rule is equivalent to: if [pic], [pic]

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